Related papers: A Mazur--Ulam theorem in non-Archimedean normed sp…

Masures are generalizations of Bruhat-Tits buildings. They were introduced to study Kac-Moody groups over ultrametric fields, which generalize reductive groups over the same fields. If A and A are two apartments in a building, their…

We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including…

This is a brief review article of various applications of non-Archimedean geometry, p-adic numbers and adeles in modern mathematical physics.

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space of dimension at least 3 and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of…

We prove a general form of the regularity theorem for uniformity norms, and deduce an inverse theorem for these norms which holds for a class of compact nilspaces including all compact abelian groups, and also nilmanifolds; in particular we…

The canonical generalizations of two classical norms on Besov spaces are shown to be equivalent even in the case of non-linear Besov spaces, that is, function spaces consisting of functions taking values in a metric space and equipped with…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively…

We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by…

A metric vector space is asymptotically metrically normable (AMN) if there exists a norm asymptotically isometric to the distance. We prove that AMN vector spaces are rigid in the class of metric vector spaces under asymptotically isometric…

In this text we prove that if X is a reduced non-archimedean analytic space and f is a analytic function on a dense Zariski-open subspace of X whose zero-locus is closed in X, then f is a meromorphic function on X. As a corollary, we deduce…

We give a different and possibly more accessible proof of a general Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(\mathbb{Z}/2)^k$-equivariant maps from a product of…

In this note it is shown that Berwald spaces admitting the same norm-preserving torsion-free affine connection have the same (weighted) Ricci curvatures. Combing this with Szab\'o's Berwald metrization theorem one can apply the…

The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach's famous book in 1932, and known as the Mazur rotations problem. Although the problem seems very difficult and rather abstract, its…

Our aim in this paper is to prove some interesting fixed point theorems for the class of asymptotically $T$-regular mappings in the framework of preordered modular G-metric spaces. Our results are novel and generalizes several know results.…

Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a real linear surjective isometry between the whole spaces. In this paper, we give an affirmative answer to…

In this article we show that positive surjective isometries between symmetric spaces associated with semi-finite von Neumann algebras are projection disjointness preserving if they are finiteness preserving. This is subsequently used to…

This Thesis is devoted to the study of Metric-Affine Theories of Gravity and Applications to Cosmology. The thesis is organized as follows. In the first Chapter we define the various geometrical quantities that characterize a non-Riemannian…

We prove the continuous non-Archimedean (resp. p-adic) Banach space and Hilbert space versions of non-Archimedean (resp. p-adic) Welch bounds proved by M. Krishna. We formulate continuous non-Archimedean and p-adic functional Zauner…

We prove a bumpy metric theorem in the sense of Ma\~{n}e for non-convex Hamiltonians that are satisfying a certain geometric property.

We introduce and study an axiomatic theory of $V$-normed $U$-modules, where $V$ is a Riesz space and $U$ is an $f$-algebra; the spaces $U$ and $V$ also have some additional structure and are required to satisfy a compatibility condition.…